In *Believing the axioms* (I and II), Penelope Maddy proposes five "rules of thumb" that she then uses to justify large cardinal axioms in set theory. These extrinsic rules are modeled after the development of set theory and the techniques of natural science. As such, applications of these rules should be found in all branches of mathematics. The most natural context for these to manifest themselves is through conjectures that are obtained by applying one of these rules of thumb in some context. I would like to hear about such conjectures (open or closed, big or small, true or false) in your area.

Maddy's five rules of thumb are:

**Maximize**: This is the opposite of Occam's Razor. The idea is that the universe should be as large as possible, anything that is likely to occur should actually occur.**Inexhaustibility**: This is the idea that the universe is too rich to be generated by a handful of basic building blocks: there should be transcendental objects.**Whimsical identity**: An object is unlikely to be the unique object satisfying a property that does not directly pertain to the object in question.**Uniformity**: The richness of the universe should not localize in one particular part, similar richness should be found in all suitably large parts.**Reflection**: If there is one object with a given property then there must also be a small (or otherwise simple) object with that property.

The above brief descriptions are mine. These were formulated by Maddy in the set-theoretic context, I attempted to phrase them in a way that would make sense in a lot of other contexts. Interpret them loosely: *object*, *property*, *universe* can be anything you want.

Note that these rules of thumb are not always good ideas and their negatives are sometimes plausible too. Although I am mainly looking for conjectures formulated in the positive sense, I think negative conjectures are also acceptable if the main reason to disbelieve the conjecture is one of the five rules above. For example, I think the Poincaré Conjecture can be understood as a negative example of the *whimsical identity* rule.

Standard Big List rules apply... One example per answer please! Try to include some brief context for the benefit of people outside your area.